Subtopic: Higher order linear ODEs with constant coefficients
Question Statement:
The general solution to the ordinary differential equation $\frac{d^2y}{dx^2} = \cos 2x$ is:
Options:
(A) $y = -\frac{\cos 2x}{4} + C_1 x + C_2$
(B) $y = \frac{\sin 2x}{2} + C_1$
(C) $y = \frac{\cos 2x}{4} + C_1 x + C_2$
(D) $y = -\frac{\sin 2x}{2} + C_1$
Official Answer Key:
A
Step-by-Step Solution & Explanation
Integrate once:
dy/dx = ∫cos 2x dx = (sin 2x)/2 + C₁
Integrate again:
y = ∫[(sin 2x)/2 + C₁] dx = −(cos 2x)/4 + C₁x + C₂
A second-order ODE must produce two arbitrary constants — this alone eliminates (B) and (D). The sign of the second integration of cos gives −cos, eliminating (C).
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